A #8 may work fine for pinks, depending on the rod. I have used an older Sage #8 SP for pinks though I prefer to use a #6 SH rod. Neither is a #7 too light for any of the other things clarkii has mentioned. So you can go with either. Most people prefer a lighter rod for coho and pinks but an 8 will help you cope with bigger rugged fish like chum.
FWIW I have caught a number of coho and steelhead on 6wt and even 5wt rods. Rod weight selection is less about species and fish size as about tackle, conditions, size of the flies, tippet size etc.
If you are fairly new to FF you might want to avoid very stiff, fast rods designed to cast a considerable distance in adverse (ie windy) conditions since you won't likely be casting that far for a little while. Rods that are a bit slower are generally easier for beginners and many experienced fly anglers prefer rods that flex into the mid and even butt sections of the rod.
There a lots of good rods now for under $200. The Echo line seems to be admired by many but there are others.
Since that is the case and you want to do some beach fishing and have $700 to spend, my thinking was put some money into a quality reel that is designed with saltwater in mind. Good reels are pricey, rods less so. Upgrades to a better rod can be done in the future and for not a whole lot of cash.
However clarkii's suggestions make just as much sense.